# Roughness

Roughness is the degree of unevenness of the surface of a solid below the scale of magnitude of its shape or undulation but above the irregularity of crystal lattice structures. The degree of roughness affects the wettability of a solid.

## Roughness parameters

Apart from wettability, many other processes and material properties are affected by the roughness, for example friction or the flow through pipes. The characterization of surfaces with regard to their roughness is therefore of great technical importance. Parameters for the roughness of a solid include:

- The
*mean roughness*, also average roughness value R_{a}. This specifies the mean distance of the actual surface height from the mean contour line. In doing so, the mean line is situated such that the sum of the height deviations assumes a minimum value. - The
*root mean square roughness*R_{q}. This is equal to the square root of the sum of the squares of the deviations from the mean contour line. - The
*roughness coefficient*r’ (see below). This specifies the ratio of the size of the overall surface to the surface projected geometrically onto a plane and is of particular importance for wettability.

## Roughness measurement

Results parameters for roughness and suitable measuring methods are defined in Norm ISO 25178. To be highlighted in this regard are non-destructive measurements by means of contactless, optical processes, for example white light interferometry or confocal microscopy. Other methods use an air current, for example, or vacuum, and utilize the gas permeability of the contact between a smooth and a rough surface.

## Roughness and wettability

As roughness goes hand-in-hand with an enlarged surface area, it affects the wettability of a solid, the contact angle (CA) of a liquid, and the adhesion. Whether the roughness increases or decreases wettability depends on the degree of wettability of the smooth material. The following observations assume that the scale of magnitude of the roughness is significantly less than the drop size of the liquid.

*Effect of roughness on wettable surfaces (CA <90°)*

A solid is wettable by a given liquid when a contact angle of less than 90° is formed. The contact angle is smaller on a rough surface of the same material. The material is therefore even more wettable. In this case, the formation of an interface between liquid and solid is energetically favorable so that the enlargement of the surface has a positive effect on the wetting.

This effect is put to practical use every day when, for example, components are ground before painting or gluing. In doing so, the bonding of adhesive and coating substances is increased in the same operation. For surface tension measurements using the Wilhelmy plate method, the platinum plate is roughened in order to optimize wetting by the sample.

According to Wenzel, the relationship between roughness and contact angle is as follows:

* cosθ* = r'cosθ *

*θ** = Measured (apparent) contact angle; *θ* = Contact angle with the smooth surface; r’ = Roughness coefficient (see above)

*Effect of roughness on non-wettable surfaces (CA >90°)*

If a liquid forms contact angles of greater than 90° on a smooth solid, the solid is not wettable. The contact angle is larger on a rough surface of the same material.

This fact is reflected in the above-mentioned equation according to Wenzel. As cos θ is negative for contact angles greater than 90°, according to this equation the apparent contact angle is larger. However, the Wenzel equation can only be applied to a limited extent as it cannot be assumed that, with little wetting, all capillaries within a rough surface are filled with liquid. Cassie and Baxter have formulated the following relationship for a situation in which only parts of the overall surface are in contact with the liquid:

* cosθ* = **r' f **cosθ + f -1 (*f = Proportion of the actually wetted surface)

*Basic states of wetting according to Wenzel and Cassie-Baxter*

This wetting behavior is known as the lotus effect, as the lotus leaf is one of the least wettable non-technical surfaces. Extremely low wettability, also referred to as superhydrophobicity, occurs as a result of the rough surface texture of a material with low surface free energy. There are many industrial applications for superhydrophobic materials or coatings. Examples include protection against fouling and water as well as self-cleaning textiles and building walls.

## Bibliography

R. N. Wenzel, Resistance of Solid Surfaces to Wetting by Water. In: Ind. Eng. Chem. 28, Nr. 8, 1936, S. 988–994

A.B. D. Cassie, S. Baxter, Wettability of Porous Surfaces. In: Trans. Faraday Soc. 40, 1944, S. 546–551

### Glossary

- Pendant drop
- Polar part
- Polynomial method
- Receding angle
- Ring tear-off method
- Rod method
- Roll-off angle
- Ross-Miles method
- Roughness
- Sessile drop
- Spinning drop tensiometer
- Spreading
- Spreading coefficient
- Stalagmometer
- Static contact angle
- Static surface tension
- Surface-active
- Surface age
- Surface excess concentration
- Surface free energy
- Surface tension
- Surfactant